Multiview geometry is the study of
two-dimensional images of three-dimensional scenes, a foundational subject in computer vision.
We determine a universal GrÃ¶bner basis for the multiview ideal of $n$ generic cameras.
As the cameras move, the multiview varieties vary in a family of dimension $11n-15$.
This family is the distinguished component of a multigraded Hilbert scheme
with a unique Borel-fixed point.
We present a combinatorial study
of ideals lying on that Hilbert scheme.

We prove that if a finite algebra $\m a$ generates a congruence
distributive variety, then the subalgebras of the powers of $\m a$
satisfy a certain kind of intersection property that fails for
finite idempotent algebras that locally exhibit affine or unary
behaviour. We demonstrate a connection between this property and the
constraint satisfaction problem.

Cubical sets and their homology have been
used in dynamical systems as well as in digital imaging. We take a
fresh look at this topic, following Zariski ideas from
algebraic geometry. The cubical topology is defined to be a
topology in $\R^d$ in which a set is closed if and only if it is
cubical. This concept is a convenient frame for describing a
variety of important features of cubical sets. Separation axioms
which, in general, are not satisfied here, characterize exactly
those pairs of points which we want to distinguish. The noetherian
property guarantees the correctness of the algorithms. Moreover, maps
between cubical sets which are continuous and closed with respect
to the cubical topology are precisely those for whom the homology
map can be defined and computed without grid subdivisions. A
combinatorial version of the Vietoris-Begle theorem is derived. This theorem
plays the central role in an algorithm computing homology
of maps which are continuous
with respect to the Euclidean topology.

The method of moving frames, introduced by Elie Cartan, is a
powerful tool for the solution of various equivalence problems.
The practical implementation of Cartan's method, however, remains
challenging, despite its later significant development and
generalization. This paper presents two new variations on the Fels and
Olver algorithm, which under some conditions on the group action,
simplify a moving frame construction. In addition, the first
algorithm leads to a better understanding of invariant differential
forms on the jet bundles, while the second expresses the differential
invariants for the entire group in terms of the differential invariants
of its subgroup.

We classify all 3 letter patterns that are avoidable in the abelian
sense. A short list of four letter patterns for which abelian
avoidance is undecided is given. Using a generalization of Zimin
words we deduce some properties of $\o$-words avoiding these
patterns.